Staff: Mentor

KE is defined as ##KE=1/2 m v^2##, where ##m## is a scalar and ##v## is a vector. ##v^2## is short for ## v \cdot v## which is the dot product, an operation which takes two vectors and returns a scalar. So although ##v## is a vector ##v^2## is a scalar, and thus KE is a scalar.

KE is defined as ##KE=1/2 m v^2##, where ##m## is a scalar and ##v## is a vector. ##v^2## is short for ## v \cdot v## which is the dot product, an operation which takes two vectors and returns a scalar. So although ##v## is a vector ##v^2## is a scalar, and thus KE is a scalar.

Thank you, this was what I was looking for.

So in general we have;

scalar*scalar= scalar?

scalar*vector = vector

vector*vector = scalar.

What about work done though?

Work done = energy

So force * distance = vector * scalar? = vector

Could you also kindly tell me about how to prove mass is scalar please.

Staff: Mentor

Not always. There are two vector products. The dot product takes two vectors and gives a scalar, but the cross product takes two vectors and gives another vector. These are usually written as ##a \cdot b## and ##a \times b## respectively.

For the rest of your questions I agree with jbriggs444's answers above, particularly for Newtonian mechanics.

Staff: Mentor

Don't forget a useful property of basic arithmetic. For real numbers, even powers are always positive, odd powers can be plus or minus. Vectors, like velocity, need to have direction and thus change sign. Scalars, like temperature or speed, have no sign.

That is not physics, but it can be useful in physics. For example, ##mv^2## is always positive. It takes the same energy to accelerate a body to an eastward velocity as to a westward velocity. You can spot that instantly because the power 2 is even.

Not always. There are two vector products. The dot product takes two vectors and gives a scalar, but the cross product takes two vectors and gives another vector. These are usually written as ##a \cdot b## and ##a \times b## respectively.

For the rest of your questions I agree with jbriggs444's answers above, particularly for Newtonian mechanics.

Don't forget a useful property of basic arithmetic. For real numbers, even powers are always positive, odd powers can be plus or minus. Vectors, like velocity, need to have direction and thus change sign. Scalars, like temperature or speed, have no sign.

That is not physics, but it can be useful in physics. For example, ##mv^2## is always positive. It takes the same energy to accelerate a body to an eastward velocity as to a westward velocity. You can spot that instantly because the power 2 is even.